The Mass Moment of Inertia

The Second Moment Integral, often just called the Moment of Inertia, can be useful in engineering mechanics calculations for a number of reasons.

For example:

Each of these three types of moments of inertia can be calculated via integration or via composite parts and the parallel axis theorem. On this page we are going to focus calculating the mass moment of inertia via integration.

Moments, Angular Accelerations, and the Mass Moment of Inertia

The mass moment of inertia (I) of an object is a constant that relates the moment applied to an object and that object's angular acceleration. It can be thought of as the rotational version of mass in Newton's second law.

Newton's Second Law
The mass moment of inertia relates the moment applied to a body to that body's angular acceleration.

The mass moment of inertia is a moment integral, specifically the second, 3D, polar, mass moment integral. To see why this relates moments and angular accelerations, we start by examining a point mass on the end of a massless stick as shown below. Imagine we want to rotate the stick about the left end by applying a moment there. We want to relate the moment exerted to the angular acceleration of the stick about this point.

Moments and Angular Acceleration for a Mass on a stick
Starting with the traditional form of Newton's Second Law, we can substitute in values to find the relationship between the moment and angular acceleration for a point mass on the end of a massless stick.

To relate these two variables, we start with the traditional form of Newton's second law, stating that the force exerted on the point mass by the stick will be equal to the mass times the acceleration of the point mass (F = m a). Next we substitute in some values. The force due to the moment will be the magnitude of the moment over the moment arm (d), and the acceleration of a point around a fixed axis is the angular acceleration times the distance between the point and the axis of rotation (d). If we substitute in these values for F and a, we end up with an equation stating that the moment exerted at the end of the stick will be equal to the mass times the distance squared times the angular acceleration.

If we were to have multiple masses all connected together with massless sticks such that they form a single body as shown below, then we would need to add up all the mass times distance squared terms to relate the moment exerted on the body to the angular acceleration.

The Second Mass Moment of Intergral
Approximating a rigid body as an infinite number of infinitely small masses all connected back to the axis of rotation, we can sum all the m d squared terms with integration.

For rigid bodies with mass distributed over some volume, we can think of the body as a collection of very small point masses all connected with a stick back to the central axis of rotation. Using integration to sum up this infinite number of infinitely small masses, we end up with the second polar mass moment integral.

This moment integral for the shape, called the mass moment of inertia, relates the moment and the angular acceleration for the body about a set axis of rotation.

The Mass Moment of Inertia

Calculating the Mass Moment of Inertia via Integration

The first step in calculating the mass moment of inertia is to determine the axis of rotation. Unlike mass, the mass moment of inertia is dependent on the axis of rotation. After choosing the axis of rotation, it is helpful to draw out the shape with the axis of rotation included.

Next we will be integrating over the mass of the object. The mass at any one location will be the volume times the density, and if we have a uniform density then we can bring the density outside of the integral leaving us with a density times a volume integral.

Pulling the density out of the mass intergral

Remembering that d is measured from the axis of rotation, we need to define dV in terms of a radius (r) moving outwards from the axis of rotation. The rate of change of the volume (dV) will be the area at exactly r distance away from that axis times the rate of change of r. This will be a cylindrical area that changes with the value of r. We therefore need to define this area as an equation relating the area to the value of r. We multiply that by r squared for the second moment integral. and evaluate from the minimum radius out to the maximum radius from the axis of rotation.

The Mass Moment of Interia
The mass moment of inertia for a general shape. 

Worked Problems:

Question 1:

A thin circular disc has a mass of 6 kg and a radius of .3 meters. Determine the mass moment of inertia for the disc about the z axis.

Problem 1 Diagram

Solution:



Question 2:

Determine the mass moment of inertia about the z axis for this general cone with base radius R, height h, and mass m.

Problem 2 Diagram

Solution: